[ 26 / 425 ] ≈ 0.0612
Step-by-step explanation:
Solution:-
- First we will describe the standard deck of 52 cards in terms of black, red and face cards found in a standard deck
- Following is a table of distribution of colored and face card found in a standard deck:
Type Number of cards
1 - 10 40
Black 26
Red 26
Face 12
- The numerical cards from digit ( 1 to 10 ) are found in all 4 suits ( Clubs, Diamonds, Spades, and Hearts ). Hence, 10 x 4 = 40
- The entire deck is split in two colors ( Red and Black ) equally. So, the number of Black and Red cards are = 52 / 2 = 26 cards.
- The face cards are of three types ( King, Queen and Jack ). These three face cards are found in each of the 4 suits. Hence, Total number of face cards are = 4 * 3 = 12
- We will now consider the probabilities asscociated with each type. We will define 3 events and write down their proability as expressed:
Event ( A ): First draw is a red card.
- The probability of this event can be determined with the help of the table given above. There are a total of 26 red card in a standard deck of 52 cards. Hence,
p ( A ) = [ Number of red cards ] / [ Total cards in a deck ]
p ( A ) = [ 26 ] / [ 52 ]
p ( A ) = 1 / 2
- After we make the first draw of a red card. Our deck distribution is changed to Number of Red cards remaining = 25 and total deck now has 51 cards remaining.
- We will define the next event as:
Event ( B ): The second draw is a face card.
- The probability of this event can be determined with the help of the table given above. There are a total of 12 face cards in a standard deck of 52 cards which is now down to 51 cards. Hence,
p ( B ) = [ Number of face cards ] / [ Total cards in a deck ]
p ( B ) = [ 12 ] / [ 51 ]
p ( B ) = 4 / 17
- After we make the first draw of a face card. Our deck distribution is changed to Number of Face cards remaining = 11 and total deck now has 50 cards remaining.
- We will define the next event as:
Event ( C ): The third draw is a black card.
- The probability of this event can be determined with the help of the table given above. There are a total of 26 black cards in the deck. The total number of cards are down to 50 cards only. Therefore,
p ( C ) = [ Number of black cards ] / [ Total cards in a deck ]
p ( C ) = [ 26 ] / [ 50 ]
p ( C ) = 13 / 25
- The entire drawing process consists of 3 events which are dependent on each draw. However, for the overall event to occur i.e drawing a red card , then a face card, and then a black card. We will multiply all three outcomes as follows:
p ( T ) = p ( A ) * p ( B ) * p ( C )
p ( T ) = [ 1 / 2 ] * [ 4 / 17 ] * [ 13 / 25 ]
p ( T ) = [ 26 / 425 ] ≈ 0.0612
